Multi-Branch Tensor Network Structure for Tensor-Train Discriminant Analysis

Higher-order data with high dimensionality arise in a diverse set of application areas such as computer vision, video analytics and medical imaging. Tensors provide a natural tool for representing these types of data. Although there has been a lot of work in the area of tensor decomposition and low-rank tensor approximation, extensions to supervised learning, feature extraction and classification are still limited. Moreover, most of the existing supervised tensor learning approaches are based on the orthogonal Tucker model. However, this model has some limitations for large tensors including high memory and computational costs. In this paper, we introduce a supervised learning approach for tensor classification based on the tensor-train model. In particular, we introduce a multi-branch tensor network structure for efficient implementation of tensor-train discriminant analysis (TTDA). The proposed approach takes advantage of the flexibility of the tensor train structure to implement various computationally efficient versions of TTDA. This approach is then evaluated on image and video classification tasks with respect to computation time, storage cost and classification accuracy and is compared to both vector and tensor based discriminant analysis methods

Graph Regularized Tensor Train Decomposition

With the advances in data acquisition technology, tensor objects are collected in a variety of applications including multimedia, medical and hyperspectral imaging. As the dimensionality of tensor objects is usually very high, dimensionality reduction is an important problem. Most of the current tensor dimensionality reduction methods rely on finding low-rank linear representations using different generative models. However, it is well-known that high-dimensional data often reside in a low-dimensional manifold. Therefore, it is important to find a compact representation, which uncovers the low dimensional tensor structure while respecting the intrinsic geometry. In this paper, we propose a graph regularized tensor train (GRTT) decomposition that learns a low-rank tensor train model that preserves the local relationships between tensor samples. The proposed method is formulated as a nonconvex optimization problem on the Stiefel manifold and an efficient algorithm is proposed to solve it. The proposed method is compared to existing tensor based dimensionality reduction methods as well as tensor manifold embedding methods for unsupervised learning applications